EFFECT OF TWO EXOSYSTEM STRUCTURES ON OUTPUT REGULATION OF THE RTAC SYSTEM

International Journal of Control Theory and Computer Modeling (IJCTCM) Vol 9, No.1/2, April 2019
DOI: 10.5121/ijctcm.2019.9201 1
EFFECT OF TWO EXOSYSTEM STRUCTURES ON
OUTPUT REGULATION OF THE RTAC SYSTEM
Ovie EseOghene1
, Muazu Muhammed Bashir2
, Sikiru Tajudeen Humble2
and
Umoh Ime Jarlath2
1
Advanced Unmanned Aerial Vehicles Laboratory (AUAVL), NASRDA, Abuja, Nigeria
2Ahmadu Bello University (ABU), Zaria, Nigeria.
ABSTRACT
This paper presents results on the output regulation of a single-input multi-output (SIMO) rotational-
translational actuator (RTAC) system. The results focus primarily on stability and robustness, which are
studied in light of the presence of externally generated exogenous input signals. Two exosystem types were
investigated and tested. Obtained results answers the question of asymptotic stabilization and tracking of a
desired trajectory in the presence of a dynamic exosystem. The results confirmed the working theory of
robust stabilization using output feedback techniques, borne out of differential-geometric observer design
principles. The utilized design showed good stability results which compares favourably with existing
works on RTAC stabilization.
KEYWORDS
RTAC,Eexogenous Signals, Feedback Regulation, Asymptotic stability, Differential Geometry, Observer
1. INTRODUCTION
The RTAC system finds applicability in systems such as dual-spin stabilized spacecraft, drill-type
systems which require stabilization in the presence of moving platforms. The RTAC system has
also been used as a benchmark system for testing various control law designs. Stabilization of the
RTAC benchmark system has been done using different approaches some of which include;
output feedback techniques [1], [2], backstepping approaches such as feedback linearization [3],
input-output feedback linearization [4], integrator backstepping [5], cascade control with
coordinate changes [6], energy and entropy based hybrid framework [7] amongst others.
Output regulation-based control is defined by tracking and stabilization goals [8] [9]. This control
paradigm targets the convergence of all or selected system states of the system towards a suitably
defined equilibrium (zero or otherwise) as time goes to infinity [10] [11]. Some of the standard
techniques which exist for output regulated systems include the use of observers as internal
model, application of separation principle and certainty principle [8] [12]. Usually, the
specification on the system to be regulated is made in presence of external disturbances and other
uncertainties which the output regulation technique defines by considering and incorporating an
external system (exosystem) to be the generator of these external signals (references and
disturbances) to be tracked or rejected. The problem can further be cast as either a reference input
tracking or disturbance input rejection problem [8] [13] [14].
In this paper, consideration has been made of two different exosystem structures, such as constant
and oscillator type signal generators. The exosystem signals are of either reference or disturbance
inputs which are modelled as part of the system and should be tracked or rejected in accordance
with the problem specification being either a disturbance rejection problem or a tracking problem

2
[13][15]. Exosystems could also in practice be known or unknown, they could also have constant
or oscillator type signal profiles [16]. This variability in types of potentially realistic disturbance
and reference signal profiles, add to the difficulty in developing nonlinear output feedback
controllers for dynamic systems [17][18].
This difficulty is encountered as part of the solution process for rejecting the exogenous signal
disturbances or tracking the reference signals from the exosystem in practice but in theory this
difficulty is in finding a closed form solution to the regulator (or Francis-Byrnes-Isidori)
equations [12] [19] for different system structures such as linear or nonlinear, square or
nonsquare, minimum or non-minimum phase, regular or non-regular [14] [20].
This work considers the RTAC system as a strictly SIMO model in all simulation experiments.
Focus is placed on the stabilization via output feedback regulation of the nonlinear RTAC
dynamic system that remains an actively researched benchmark system within the control
community. The results for the two tested exosystem structures is also presented.
The organization of the rest of the paper is as follows: Section 2 introduces the RTAC system and
the mathematical model, section 3 summarizes the output feedback problem for nonlinear system.
Section 4 addresses the exosystem structure, section 5 treats the output regulator in detail, section
6 touches on the stable regulator or controller design, section 7 presents the important results and
section 8 concludes.
2. THE RTAC SYSTEM
Practical research interest in the RTAC system stems from the fact that it is an underactuated
coupled nonlinear system with four states and a single input. This paper treats the RTAC as a
SIMO system under observer based output regulation. It focuses on the internal model component
and observer estimator. The main contribution was in the regulator formulation without squaring
the plant. Figure 1 shows the structure of the RTAC and some of its parameters.
Fig. 1: RTAC System
The pendulum is the actuated member while the proof mass is driven forward and backward. This
pendulum position is determined by the designed controller while the proof mass is placed at
specific positions with the influence of disturbances on the proof mass taken into account.
2.1. RTAC EQUATIONS
The nonlinear RTAC model used by [21] was modified in [22] as follows

3
2
2
( )q sin cos
( ) cos
M m kq ml ml F
I ml ml q N
 
 
    
  
(1)
where M, is the translation or proof mass, m, the mass of pendulum bob, l, the length of the
pendulum arm, I, mass moment of inertia of the pendulum arm, g, acceleration due to gravity, θ,
the displacement angle from the vertical and x, is the translational displacement of M. The
nonlinear equation (2) was rearranged into the following form:
1 2
2
2 4 3 32 1 2
2
3 4
2
1 3 2 2 4 3 3 31
4
sin cos
cos sin cos cos
nl nl nl nl
nl nl nl nl
x x
mlM x x mlN xM kx M F
x
D D D D
x x
mlkx x m l x x x mlF xM N
x
D D D D

    

   
(2)
Subsequent linearization of the above nonlinear equation was used for the initial analysis and
design of the output feedback regulator.
2 2
1
M kq M FmlN
q
Dl Dl Dl
M Nmlkq mlF
Dl Dl Dl

 
  
 
  
(3)
also expressed in state space form as
1 2
32 1 2
2
3 4
1 1
4
nl nl nl
nl nl nl
x x
mlNxM kx M F
x
D D D
x x
mlkx M N mlF
x
D D D

   

  
(4)
The linearization was necessary to engage in any constructive development of the output
feedback regulator. The working matrices were obtained from the RTAC structure in (4).
3. OUTPUT REGULATION SETUP
As presented in [23], the generalized nonlinear system for output regulation was given by
( , , )
( )
( , , )
x f x u w
w s w
y h x u w



(5)
with analytic, input affine form given as
( ) ( ) ( )
( )
( ) ( )
x f x g x u g x w
w s w
y h x q w
  

 
(6)

4
A particular linear equivalent of the setup in equation (6) can be taken as presented by [24]
u wx Ax B u P v
v Sv
y Cx Du Qv
  

  
(7)
Equation (7) represent plant dynamics, exosystem and output respectively whose working
parameters discernible from equation (4). The variables x, u, v in equation (7) are in general
vector valued with sizes x ϵ Rn
, u ϵ Rm
and v ϵ Rq
. While the matrices A ϵ Rnxn
, B ϵ Rnxm
, P ϵ Rnxq
, S ϵ Rqxq
, C ϵ Rpxn
, D ϵ Rpxm
, Q ϵ Rpxq
. Here the exosystem plays the role of a signal generator,
which encapsulates the possible class of reference and disturbance signals to be tracked or
rejected by the system. In [24] the class of reference and disturbance signals generated with the
different initial states of the exosystem were shown to depend on the operator S. This dependence
of the system response on the exosystem structure will be explicitly shown in later part of this
paper through analysis and results.
3.1 OUTPUT FEEDBACK REGULATOR FOR THE RTAC
The RTAC analysis utilized the following parameters taken from Sun et al., (2016) [9]
Table 1: Table of Parameters for the CIP.
Parameter Value Unit
Proof mass (M) 3.82 Kg
Pendulum mass (m) 0.5 Kg
Pendulum length(l) 0.12 m
Mass MOI (I) 0.0003186 Kg/m2
Spring constant (ks) 427
Acc. due Gravity (g) 9.81 m/s2
3.2. STATE SPACE REALIZATION OF THE RTAC
From the linear system given by equation (7), the working matrices for the values in Table 1 are
derived as follows:
0 1 0 0
111.167 0 0 0
0 0 0 1
887.347 0 0 0
A
 
 
 
 
 
 
,
0
2.0781
0
149.623
B
 
 
 
 
 
 
 1 0 1 1C 
The uncertain disturbance term on the internal state dynamics and the output is captured by the
matrices P and Q respectively

5
0.649 0.8456
1.1812 0.5727
0.7585 0.5587
1.1096 0.1784
P
  
 
 
  
 
 
,
0.1969 0.8003
0.5864 1.5094
0.8519 0.8759
Q
 
 
  
  
the disturbance P and Q are selected to be random noise terms with the requirement being that
they must not be zero or sparse so as to avoid singular values or not-a-number (NaN) errors
during simulation.
4. EXOSYSTEM SELECTION
The exosystem generates the reference signals and disturbances fed into the system. Certain
conditions must however be satisfied for the exosystem to be acceptable as a signal generator for
the output regulated setup. These conditions have been extensively discussed in the work of [24].
Generally, the selected exosystem must be antistable. One specific example of this antistable
behaviour means the exosystem satisfies the Poisson stability criterion [10] [11]. However, other
types of exosystem structures have been considered and used such as anti-Hurwitz stable (having
non-negative real part for its eigenvalue) [25], non-negative real part with semi-simple map [26].
An often utilized form of the exosystem employs a linear time invariant (LTI) dynamic system
[27]
0; (0)
ref
v Sv v v W
w Ev
y Fv
  

 
(8)
Where v is the dynamic ODE representing the exosystem, w is the disturbance signal generated
from the exosystem dynamic and yref is the reference signal generated accordingly. The system in
(8) generates the reference signals and disturbances and have S, E and F appropriately sized. The
exosystem map S, satisfies the condition given byRe( (S)) = 0 , being read as; the spectrum of
the matrix S having all real parts equals zero. Implying, all eigenvalues of the chosen map S, must
be Poisson stable or have their occurrence on the imaginary axis of the complex plane.
Although linear exosytems structures such as equation (8) are easier to work with and more
tractable in solving the full system regulator equations [11], practical occurrences of the
exosystem demand that nonlinear structures be also considered [28]. Also considered in certain
implementations of the regulation problem is the availability or observability of the exosystem
states under constrained conditions. The observed states usually used in feedforward arrangement
[9] [25]. The current treatment has however investigated linear time invariant exosystem
structures.
This work tested two different exosystem structures. The first is a constant-type disturbance
generator, the second was a 2x2 dimension disturbance generator model. The first exosystem
candidate selected was 0w 
1 1
2 2
0 0
0 0
w w
w w
    
    
    
(9)

6
Here
0 0
0 0
exoS
 
  
 
, which has a constant matrix value.
Another exosystem candidate has the oscillator of the form
1 2
2 1
w w
w w
 
 
 
(10)
With
0 1
1 0
exoS
 
  
 
. In both cases, the states 1w and 2w can be used for the disturbance and
reference signal profiles respectively to be injected into (8).
5. OUTPUT FEEDBACK REGULATOR ANALYSIS
A regulator was constructed having the output error feedback form. The synthesis requires the
construction of a compensator of the form
1 2z G z G e
u Kz
 

(11)
Where K = [Kx Kv]. The following set of parameters were designed for Kx, Kv, L1, L2, G1, G2, so
that the following conditions hold:
i. The pair A, B is stabilizable by Kx s.t. is Hurwitz
ii. There exists a unique solution for X and U to the regulator equation pair
0
exoXS AX Bu P
CX Q
  
 
(12)
Such that the disturbance feedback gain Kv was computable as: Kv=U-KxX
iii. finally the internal model-based observer was synthesized using the parameters obtained from
the regulator equations in equations (29) and the selected minimal subset of the matrix
1 1
2 2
l
A LC P LQ
A
L C S L Q
  
  
   (13)
for a solution to L1, L2 and subsequently G1 and G2 for a complete solution to the controller
design.
iv. All the previous computations led to the closed loop system given by
2 1 2
0 0
xv
clsys
exo
A BK P
A G C G G Q
S
 
 
  
 
  (14)
, from which we derived the subset matrices

7
2 1
xv
cllp
A BK
A
G C G
 
  
  (15)
and Sexo (for any selected exosystem structure) whose closed loop internal stability is deduced
by checking the Hurwitz condition of (15) such that  spec Acllp C
 and   0
exospec S C .
6. STABILITY ANALYSIS AND CONTROLLER DESIGN
Considering the RTAC system as a SIMO model with target output selected as 1 3 4, ,( )x x xy  ,
suitable generalized scalar Lyapunov function for the chosen outputs becomes
1 1 3 3 4 4V(x)= x Px +x Px x PxT T T
 (16)
satisfying the Lyapunov equation PA+AT
P+I=0, where P is a positive definite and symmetric
(PDS) matrix, I is the identity matrix of suitable size and A, the stable transmission matrix from
linearization of (2). However the RTAC system is considered a SIMO model with more specific
Lyapunov function given by
2 2 2
1 1 3 3 4 4
1 1 1
( , ) ( ) ( ) ( )
2 2 2
r r rV x t x x x x x x      (17)
Equation (17) met the following conditions for Lyapunov function construction in continuous
time non-autonomous systems (CTNAS):
1. 0V(0,t ) = 0
2. 0V(x, t) > 0 x 0 D, t t   
3. there exists class K functions α(.), β(.) and γ(.) which satisfy the inequality
     , , 0|| || || ||x V x t x t t    
4. and finally  ,V x t satisfies the inequality
V(x,t) - (||x||) < 0, t t0  
Equation (17) satisfied conditions (1)-(4) and satisfactorily generated desired control signals.
7. EXPERIMENTS AND RESULTS
The computed feedback gain that stabilized the closed loop system A BF was obtained as F=
[237.3198, 17.3309, 91e-003, 48.2e-003]. The FBI regulator equations were solved for X and U
and the following results were obtained as solution to the FBI equations with the oscillator
exosystem

8
196.8614 003 800.321 003
0.0 0.0
586.4426 003 1.509
851.887 003 875.874 003
osc
e e
X
e
e e
   
 
 
  
 
   
 oscU = -1.1519e + 000 4.7416e + 000
with the constant exosystem, U was obtained as
 1.1577 4.7359constU  
these solutions were unique and proved the existence of the regulator. The computed solutions
also made possible the computation of a unique value for the internal model gain Kv, which was
obtained as Kv = [−111.2459 452.2505] when oscillator-type exosystem was used and Kv =
[−111.2517 452.2448] when constant-type exosystem was applied. The input vector was formed
as Kxv = [Kx Kv] in both cases. Other parameters computed include the dynamic output gain L1,
L2, G1, G2, which for space constraint has been moved to the appendix. Simulated experiments
were made to show the uncompensated, compensated and combined responses of the RTAC
under initial perturbation.
Starting with an initial condition, IC = (0.5 0 0.75 0), the unforced closed loop response was
unstable. This necessitated development of a feedback regulator.
Fig. 2: Compensated Initial Response for IC = (0.5 0 0.75 0)
Fig. 2, shows the compensated response with the feedback and output gains acting to make the
internal model and plant stable.

9
Fig. 3: Combined Initial Response for IC = (0.5 0 0.75 0)
Fig. 3, shows the combined compensated response with the internal model acting and all states
stabilizing under 1.175sec.
For IC = (1 0 − 0.5 0), a second experiment was made to test for internal stability of the designed
controller.
Fig. 4: Compensated Initial Response for IC = (1 0 − 0.5 0)
Fig. 4 gives the initial response when the states were perturbed by (1.0m 0 π/4rad 0rad/s).

10
Fig. 5: Combined Initial Response for IC = (1.0m 0 π/4rad 0rad/s).
Fig. 5 shows the combined plot of the initial perturbation response of the RTAC system showing
relative settling times of the various states.
Fig. 6: Compensated Initial Response for IC = (1.0m 0 π/4rad 0rad/s)
Fig. 6 gives the initial response when the states were perturbed by (1.0m 0 π/4rad 0rad/s)
Fig. 8: Angular Position Response for Two Exosystem Types

11
The effect of selecting either the oscillator or constant type exosystem is viewed through the
angular response when an initial perturbation was applied. The oscillator type exosystem showed
a slowly decaying behavior compared to the constant type exosystem response which converged
faster from within a local region of the origin. Fig. 8, shows the angular response of the RTAC
system from application of the two types of exosystem structures. The constant exosystem
showed faster convergence to the equilibrium in 2.5sec. While the oscillator-type exosystem had
a slow convergence profile which only settled to the equilibrium after 45sec in the experiments
conducted.
8. CONCLUSION
The RTAC benchmark control system has been experimented with in simulation. An output
feedback regulator stabilizer was synthesized and put in place to stabilize the system internal
dynamics. It was seen that from any starting condition, the system settles down to its stable
equilibrium within 1.175sec from start of the simulation. This stability has been achieved with the
aid of carefully designed gains which utilized both pole placement and polynomial matching for
its synthesis. The system was therefore made stable by output regulation. The effect of a constant
and oscillator-type exosystem were also considered. The constant type exosystem in equation (9)
settled down faster to its equilibrium in 2.5sec while the oscillator-type exosystem described in
equation (10) settled down after 45sec. However both attained asymptotic stability in their
response profiles.
REFERENCES
[1] Z. P. Jiang, D. J. Hill and Y. Guo (1998). “Stabilization and tracking via output feedback for the
nonlinear benchmark system.” Automatica
[2] G. Escobar, R. Ortega, H. Sira-Ramirez (1999). “Output-Feedback Global Stabilization of a
Nonlinear Benchmark System Using a Saturated Passivity-Based Controller.” IEEE Transactions on
Control Systems Technology, Vol. 7, No. 2, March.
[3] Z. P. Jiang and I. Kanellakopoulos (2000). “Global Output-Feedback Tracking for a Benchmark
Nonlinear System.” IEEE Transactions On Automatic Control Vol. 45, No. 5, pp 1023-1027, May.
[4] A. Oveisi, M. M. Mohammadi, M. Gudarzi, S. M. H. Hasheminejad (2012). “Tracking Controller
Design and Analysis for RTAC Using Input/Output Feedback Linearization Approach.” 4th Iranian
Conference on Electrical and Electronics Engineering (ICEEE2012) Islamic Azad University,
Gonabad-Branch 28th-30th August.
[5] R. T. Bupp, D. S. Bernstein, V. T. Coppola (1995). “A Benchmark Problem for Nonlinear Control
Design: Problem Statement, Experimental Testbed, and Passive Nonlinear Compensation.”
Proceedings of the American Control Conference, Seattle Washington. June.
[6] X. Wu, and X. He (2016). “Cascade-Based Control of a Benchmark System.” 28th Chinese Control
and Decision Conference (CCDC).
[7] J. M. Avis, S. G. Nersesov, R. Nathan, H. Ashrafiuon, K. R. Muske (2010). “A comparison study of
nonlinear control techniques for the RTAC system.” Nonlinear Analysis: Real World Applications
11, pp 2647-2658.
[8] M. Guo and L. Liu (2016). “Output regulation of output feedback systems with unknown exosystem
and unknown high-frequency gain sign.” Transactions of the Institute of Measurement and Control
1–8. DOI: 10.1177/0142331216654535
[9] H. Basu and S. Y. Yoon (2018). “Robust Cooperative Output Regulation under Exosystem
Detectability Constraint.” International Journal of Control, DOI:10.1080/00207179.2018.1493537

12
[10] M. Bando and A. Ichikawa (ND). “Adaptive Output Regulation of Nonlinear Systems Described By
Multiple Linear Models.” Kyoto University, Department of Aeronautics and Astronautics,
Yoshidahonmachi, Sakyo-ku, Kyoto, 606-8501, Japan
[11] V. Sundarapandian (ND). “A Relation between Output Regulation and Observer Design for
Nonlinear systems.”
[12] L. Wang, A. Isidori, H. Su, W. Xu (2014).” Partial-State Observer for a Class of MIMO Nonlinear
Systems.” Preprints of the 19th World Congress. The International Federation of Automatic
Control, Cape Town, South Africa. August 24-29, 2014
[13] A. Serrani (2005). “Robust Semi global Nonlinear Output Regulation: The Case of Systems in
Triangular Form. Output Regulation of Nonlinear Systems in the Large. ”CeSOS-NTNU 2005
[14] A. Isidori (2011). “The Zero Dynamics of a Nonlinear System: from the Origin to the Latest
Progresses of a Long Successful Story.”Proceedings of the 30th Chinese Control Conference July
22-24, 2011, Yantai, China.
[15] A. Serrani (2012).” Output regulation for over-actuated linear systems via inverse model allocation.”
51st IEEE Conference on Decision and Control December 10- 13, 2012.Maui, Hawaii, USA
[16] W. Sun, J. Lan, J. T. W. Yeow (2013). “Constraint adaptive output regulation of output feedback
systems with application to electrostatic torsional micro mirror.” International Journal of Robust
and Nonlinear Control. Published online in Wiley Online Library (wileyonlinelibrary.com). DOI:
10.1002/rnc.3100
[17] A. Astolfi, D. Nesic, A. R. Teel (2008). “Trends in Nonlinear Control.” Proceedings of the 47th
IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11.
[18] Y. Su and J. Huang (2012). “Cooperative Output Regulation of Linear Multi-agent Systems.” IEEE
Transactions on Automatic Control, 57(4)
[19] Z. Chen and J. Huang (2004). “Robust Output Regulation with Nonlinear Exosystems. “Proceedings
of the 2004 American control conference. Boston Massachusetts. June 20-July 2nd
2004
[20] J. Chang, C. Wua, C. Shih (2015). “Output feedback high-gain proportional integral control for
minimum phase uncertain systems.” Journal of the Franklin Institute 352 (2015) 2314-2328
[21] N. Sun, Y. Wu, Y. Fang, H. Chen, B. Lu (2016). “Nonlinear continuous global stabilization control
for underactuated RTAC systems: Design, analysis, and experimentation.” IEEE/ASME
Transactions on Mechatronics October.
[22] E. Ovie (2018). “Development of a Dynamic Output-Feedback Regulator for Stabilization and
Tracking of Nonsquare Multi-Input Multi-Output Systems.” A Thesis Submitted to the School of
Postgraduate Studies, Ahmadu Bello University, and Zaria. March.
[23] C. O. Aguilar and A. J. Krener (2012). “Patchy solution of a Francis–Byrnes–Isidori partial
differential equation.” International Journal of Robust and Nonlinear Control. Published online in
Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.2776
[24] L. Paunonen (2013). “The Role of exosystems in Output Regulation.” IEEE Transactions On
Automatic Control pp. 1-6, 2013.
[25] R. Schmid, L. Ntogramatzidis, S. Gao. (2013). “Nonovershooting multivariable tracking control for
time- varying references.” 52nd IEEE Conference on Decision and Control December 10-13, 2013.
Florence, Italy

13
[26] D. Carnevale, S. Galeani, M. Sassano (2016). “Data-Driven Output Regulation by External Models
of Linear Hybrid Systems with Periodic Jumps.” 2016 IEEE 55th Conference on Decision and
Control (CDC) ARIA Resort & Casino December 12-14, 2016, Las Vegas, USA
[27] Paunonen, L. (2016). “Controller Design for Robust Output Regulation of Regular Linear Systems.”
IEEE Transactions on Automatic Control, 61(10), 2974-2986. DOI: 0.1109/TAC.2015.2509439
[28] L. Wang and C. M. Kellett (2019).”Adaptive Semiglobal Nonlinear Output Regulation: An
Extended-State Observer Approach.”
AUTHOR
Ovie EseOghene is a PhD candidate of Control Engineering at the Ahma du Bello
University Zaria, Nigeria. His research is on output Regulation of nonsquare
nonlinear systems. He is currently with The Advanced Unmanned Vehicles
laboratory, Ebonyi State, Nigeria.
Muhammed Bashir Muazu is a Professor of Evolutionary Algorithms at the
Department of Computer Engineering, Ahmadu Bello University, Zaria
Dr. Tajudeen Humble Sikiru has a Phd in Powersystems and Works with the
Ahmadu Bello University, Zaria. He is also the Managing Director at Rana World
Tech Shop Limited
APPENDIX A: INTERNAL MODEL GAINS
The Luenberger observer parameters for the oscillator-type exosystem experiment is as given by
the variables G1 and G2
168 1 0 0 32.424 135.3
8172.2 105.96 2.139 0.777 1633.7 6637.2
0 0 49 0 28.642 72.53
1
83892.7 7628.8 154.01 77.041 16.36 3 66840.6
15.35 0 11.46 34.913 20 22.65 15
35.45 0 41.67 35.71 3.11 15 60
G
e
e
e
  
 
     
  
  
  
     
      

168 0 0
7.12 3 0 0
0 49 1
2
887.35 0 133
15.35 11.46 34.912
35.45 41.674 35.71
e
G
 
 
 
 
  
 
 
    
While the corresponding Luenberger observer parameters for the constant-type exosystem
experiment is as given by the variables G1c and G2c

14
168 1 0 0 32.424 135.3
8172 105.96 2.14 777.21 3 1.634 3 6.64 3
0 0 49 0 28.64 72.53
83.893 3 7.63 3 154.01 77.04 16.36 66.84
1
400 0 0 0 78.745 320.13
0 0 0 0 0 0
0 0 144 0 84.45 217.35
0 0 0 61.42 52.323 53.796
e e e
e e
G c
  

     
  

 
  


  
   










168 0 0
7118.2 0 0
0 49 1
887.35 0 133
2
400 0 0
0 0 0
0 144 0
0 0 61.42
G c
 
 
 
 
 
 
 
 
 
 
  
 

EFFECT OF TWO EXOSYSTEM STRUCTURES ON OUTPUT REGULATION OF THE RTAC SYSTEM

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